Eigenvalues and Markov Chains

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1 Eigenvalues and Markov Chains Will Perkins April 15, 2013

2 The Metropolis Algorithm Say we want to sample from a different distribution, not necessarily uniform. Can we change the transition rates in such a way that our desired distribution is stationary? Amazingly, yes. Say we have a distribution π over X so that π(x) = w(x) y X w(y) I.e. we know the proportions but not the normalizing constant (and X is much too big to compute it).

3 The Metropolis Algorithm Metropolis-Hastings Algorithm 1 Create a graph structure on X so the graph is connected and has maximum degree D. 2 Define the following transition probabilities: 1 p(x, y) = 1 2D (max{w(y)/w(x), 1}) if x and y are neighbors. 2 p(x, y) = 0 if x and y are not neighbors 3 p(x, x) = 1 y x p(x, y) 3 Check that this Markov chain is irreducible, aperiodic, reversible and has stationary distribution π.

4 Example Say we want to sample large independent sets from a graph G. I.e. P(I ) = λ I where Z = J λ J where the sum is over all independent sets. Note that this distribution gives more weight to the largest independent sets. Use the Metropolis Algorithm to find a Markov Chain with this distribution as the stationary distribution. Z

5 Linear Algebra Recall some facts from linear algebra: If A is a real symmetric, n n matrix, then A has real eigenvalues and there exists an orthonormal basis of R n consisting of eigenvectors of A. The eigenvalues of A n are the eigenvalues of A raised to the n Rayleigh Quotient form of eigenvalues

6 Perron-Frobenius Theorem Theorem Let A > 0 be a matrix with all positive entries. Then there exists an eigenvalue λ 0 > 0 with eigenvector x 0 all of whose entries are positive so that 1 If λ λ 0 is another eigenvalue of A then λ < λ 0. 2 λ 0 has algebraic and geometric multiplicity 1

7 Perron-Frobenius Theorem Proof: Define a set of real numbers Λ = {λ : Ax λx for some x 0}. Show that Λ [0, M] for some M. Then let λ 0 = max Λ. From the definition of Λ, there exists an x 0 0 so that Ax 0 λ 0 x 0. Suppose Ax 0 λx 0. Then let y = Ax 0 and A(y λ 0 x 0 ) = Ay λ 0 y > 0 since A > 0. But this is a contradiction. So Ax 0 = λ 0 x 0.

8 Perron-Frobenius Theorem Now pick an eigenvalue λ λ 0 with eigenvector x. Then and so λ λ 0. A x Ax = λx = λ x Finally, we show that there is no other eigenvalue λ = λ 0. Consider A δ = A = δi for small enough δ so the matrix is still positive. A δ has eigenvalues λ 0 δ and λ δ, and λ 0 δ λ δ. But if λ λ 0 is on the same circle in the complex plane as λ 0, this is a contradiction. [picture]

9 Perron-Frobenius Theorem Finally, we address the multiplicity. Say x and y are linearly independent eigenvectors with eigenvalue λ 0. Then find α so that x + α y has non-negative entries, but at least one 0 entry. But since A > 0 and A(x + αy) = λ(x + αy) there is a contradiction.

10 Application to Markov Chains Check: the conclusions of the Perrron-Frobenius theorem hold for the transition matrix of a finite, aperiodic, irreducible Markov chain.

11 Rate of Convergence Theorem Consider the transition matrix P of a symmetric, aperiodic, irreducible Markov Chain on n states. Let µ be the uniform (stationary) distribution. Let λ 1 = 1 be the largest eigenvalue and λ 2 the second-largest in absolute values. Then π (x) m µ TV n λ 2 m Proof: Start with the Jordan Canonical form of the matrix P. (A generalization of diagonalizing - we ll assume P is diagonalizable), i.e. D = UPU 1 The rows of U are the left eigenvectors of P and the columns of U 1 are the right eigenvectors.

12 Rate of Convergence Order the eigenvalues 1 = λ 1 > λ 2 >.... The left eigenvector of λ 1 is the stationary distribution vector. The first right eigenvector is the all 1 s vector. Now write P n = U 1 D n U. Write π 0 is the eigenvector basis: π 0 = µ + c 2 u c n u n and π m = π 0 P m = µ + where λ j λ 2 < 1. n c j λ m j u j j=2

13 Eigenvalues of Graphs The adjacency matrix A of a graph G is the matrix whose i, jth entry is 1 if (i, j) E(G). The normalized adjacency matrix turns this into a stochastic matrix - for example, if G is d-regular, we divide A by d. For d-regular graph, with normalized adjancey matrix A, What is λ 1? What does A correspond to in terms of Markov Chains? What does it mean if λ 2 = 1? What does it mean if λ n = 1?

14 Cheeger s Inequality For a d-regular graph, define the edge expansion of a cut S V as: h(s) = E(S, S c ) d min{ S, S c } The edge expansion of a graph G is h(g) = min S V h(s)

15 Cheeger s Inequality Theorem (Cheeger s Inequality) Let 1 = λ 1 λ 2... be the eigenvalues of the random walk on the d-regular graph G. Then 1 λ 2 2 h(g) 2(1 λ 2 ) What does this say about mixing times of random walks on graphs?

16 Ehrenfest Urn What are the eigenvalues and eigenvectors of the Ehrenfest Urn?

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